$\begin{align}
& x\in \left[ \pi ,2\pi \right]\Rightarrow -1\le \cos x\le 1\Rightarrow -b\le b\,\cos x\le b \\
& \Rightarrow a-b\le a+b\,\cos x\le a+b \\
& \Rightarrow \left\{ \begin{matrix}
a-b=\frac{1}{2} \\
a+b=\frac{3}{2} \\
\end{matrix} \right. \\
& 2a=2\Rightarrow a=1,b=+\frac{1}{2} \\
& \Rightarrow f(x)=1+\frac{1}{2}\cos x \\
& \Rightarrow f(\frac{5\pi }{3})=1+\frac{1}{2}\cos (\frac{5\pi }{3})=1+\frac{1}{2}\cos (2\pi -\frac{\pi }{3}) \\
& =1+\frac{1}{2}\cos \frac{\pi }{3}=1+\frac{1}{2}\times \frac{1}{2}=\frac{5}{4} \\
\end{align}$